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In this work, we propose a biquaternionic reformulation of a fractional monochromatic Maxwell system. Additionally, some examples are given to illustrate how the quaternionic fractional approach emerges in linear hydrodynamics and elasticity.

The past few decades have witnessed a surge of interest in research on the theory of the Maxwell system. A technique to study the Maxwell system is to reduce it to the equivalent Helmholtz equation. In a series of recent papers, diverse applications of the Maxwell system theory have been studied (see [

The Dirac equation is an important one in mathematical physics used to represent the Maxwell system through several ways, which has attracted the attention of physicists and engineers (see [

A new approach for the study of the Maxwell system by using the quaternionic displaced Dirac operator, rather than working directly with the Helmholtz equation, appeared recently.

The quaternionic analysis gives a tool of wider applicability for the study of electromagnetic problems. In particular, a quaternionic hyperholomorphic approach to monochromatic solutions of the Maxwell system is established in [

The fractional calculus goes back to Leibniz, Liouville, Grunwald, Letnikov, and Riemann. There are many interesting books on this topic as well as in fractional differential equations (see, e.g., [

Nowadays, fractional calculus is a progressive research area [

The fractional derivative operators are nonlocal, and this property is very important because it allows modeling the dynamic of many complex processes in applied sciences and engineering (see [

Recently, Ferreira and Vieira [

The main goal of this paper is to describe the very close connection between the

After this brief introduction, let us give a description of the sections of this paper. Section 2 contains some basic and necessary facts about fractional calculus, fractional vector calculus, and the connections between quaternionic analysis and fractional calculus. In Section 3, we present some examples of fractional systems in physics. Finally, Section 4 is devoted to the study of a fractional monochromatic Maxwell system and summarizes the main achievements of this study.

In this section, we introduce the fractional derivatives and integrals necessary for our purpose and review some standard facts on fractional vector calculus and basic definitions of quaternionic analysis.

Definitions and results of fractional calculus are established in this subsection (see [

Let a real-valued function

The left Caputo fractional derivative of order

Here and subsequently,

It is easily seen that

Unfortunately, in general, the semigroup property for the composition of Caputo fractional derivatives is not true. Conditions under which the law of exponents holds are established in the next theorem, which follows the main ideas proposed in [

Let

Applying (

From [

Consequently,

But

We follow Kravchenko [

Let

Let

The multiplication of two quaternions

An

Let us denote by

If

The following first-order partial differential operator is called the Dirac operator:

Because

The Helmholtz operator

For an

The interested reader is referred to [

In past decades, there has been considerable effort in the literature to study boundary problems of pure mathematics and mathematical physics for domains with highly irregular boundaries like nonrectifiable, finite perimeter, fractals, and flat chains (see, for instance, [

In

Let

The fractional Nabla operator in coordinates

Following the ideas of [

Let

If

If

The fractional curl operator is defined by

Note that these fractional differential operators are nonlocal and depend on the

The following relation for fractional differential vector operations is easily adapted from [

A definition of the 3-parameter fractional Laplace and Dirac operators using left Caputo derivatives can be found in [

The fractional Dirac operator

We can now state (paraphrasing the Dirac operator case) the fact that the solution of the fractional Dirac operator is a fractional harmonic.

By straightforward calculation, we have

In general, physical models can be formulated using the fractional derivatives, where the kernels are interpreted as power-law densities of states, and the fractional order of the derivative corresponds to the physical dimensions of the material [

In this section, we illustrate some examples where the quaternionic fractional approach emerges in linear hydrodynamics and elasticity. These fractional physical systems are motivated by [

Let a vector field

For

Note that for

The velocity field

Under the assumption of negligible inertial and thermal effects, the time-independent velocity field

The equation (

Suppose a solid body translates with constant velocity

A 3-dimensional field

The fractional calculus can be used to establish a fractional generalization of nonlocal elasticity in two forms: the fractional gradient elasticity theory (weak nonlocality) and the fractional integral elasticity theory (strong nonlocality) (see [

Many applications of fractional calculus amount to replacing the spatial derivative in an equation with a derivative of fractional order. So, we can consider a generalization of (

Precisely, we will propose the following transformations:

Then, we get the fractional Lamé-Navier system associated with the transformations (

Combining (

Consequently, the fractional Lamé-Navier system (

Let us denote

Having in mind the conditions relating

Note that the operational equation involving (

Observe that the term

The behavior of electric fields

The relations between electric fields

Analogously, we have a nonlocal equation for the magnetic fields

A feasible way of appearance of the Caputo derivative in the classical electrodynamics can be found in [

If we have

The integration by parts now leads to

The nonlocal properties of electrodynamics can be considered as a result of dipole-dipole interactions with a fractional power-law screening that is connected with the integrodifferentiation of noninteger order (see [

Consider the kernel

Then, (

Let us apply (

The main idea behind the use of fractional differentiation, for describing real-world problems, is their abilities to describe nonlocal distributed effects. For example, a power-law long-range interaction in the 3-dimensional lattice in the continuous limit can give a fractional equation (see [

Also, the methodology used in [

The fractional Maxwell system (

We will assume that the electromagnetic characteristics of the medium do not change in time. If in addition they have the same values in each point of the cube

A monochromatic electromagnetic field has the following form:

Substituting (

The quantities

The following fractional wave equations can be found in [

Substituting (

The above fractional Helmholtz equations motivate the introduction of the fractional Helmholtz operator

Next, the fractional Helmholtz operator, can be factorized as

The formulation of (

In view of the factorization (

Applying the fractional divergence operator to the last equation in (

In order to rewrite (

Introduce the following pair of purely vector biquaternionic functions

Now, we formulate the main result of this paper, which consists of a biquaternionic reformulation of a fractional monochromatic Maxwell system.

The fractional quaternionic equations

Let

Applying

Thus, (

On the contrary, suppose that

Substituting (

From the last equality, we conclude that (

Separating the vector and scalar parts in (

The main purpose of this paper was to explore the very close connection between the 3-parameter quaternionic displaced fractional Dirac operator with a fractional monochromatic Maxwell system using Caputo derivatives. With this aim in mind, a biquaternionic reformulation of such a system was studied. Moreover, some examples to illustrate how the quaternionic fractional approach emerges in linear hydrodynamic and elasticity are given. As future works, the formulation of a fractional inframonogenic functions theory is suggested.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

All authors contributed equally to this work. All authors read and approved the final manuscript.

Juan Bory Reyes was partially supported by Instituto Politécnico Nacional in the framework of SIP programs (SIP20195662). Yudier Peña-Pérez gratefully acknowledges the financial support of the Postgraduate Study Fellowship of the Consejo Nacional de Ciencia y Tecnología (CONACYT) (grant number 744134).